Download 1. Determination of Activation Energy for Dehydroxylation of Illite

1. Determination of Activation Energy for Dehydroxylation of Illite
The kinetics of chemical reactions are commonly expressed as
da
= k f (a ) ,
dt
(1)
æ -Ea ö
k = Aexp ç
÷,
è RT ø
(2)
where  is the reacted fraction of a substance (0 ≤  ≤ 1; = 1 if the entire substance is
reacted), t is reaction time, k is reaction rate, and f( is a kinetic (rate-limited) function
determined by the reaction mechanism (this can be a first- or second-order reaction, one-, two- or
three-dimensional diffusion, two- or three-dimensional phase boundary, or two-, three-, or ndimensional nucleation [e.g., Koga, 1997]. The relationship between reaction rate k and
temperature, as expressed by the Arrhenius equation, is
where A is a constant (pre-exponential term), Ea is the activation energy necessary for a reaction
to occur, R is the gas constant (8.31447 J K-1 mol-1), and T is temperature (K).
The Friedman analysis (or differential isoconversional method) is widely used to determine
Ea independent of the kinetic function in experiments carried out at different heating rates. From
Eqs. 1 and 2,
da
E 1
(3)
= - a + ln [ Af (a )] .
dt
RT
At any fixed , the logarithmic term on the right side of Eq. 3 is a constant, so the logarithm of
the conversion rate d/dt over the inverse of temperature 1/T describes a straight line with a
ln
slope of –Ea/R. In this way, Ea can be determined independently from the selection of the kinetic
function.
Thermogravimetric profiles at heating rates of 5, 10, and 20 °C min-1 were obtained using a
Netzsch STA 449 C Jupiter balance. Using the kinetic software Netzsch Thermokinetics with Ea
values constrained by Friedman analysis on the basis of Eq. 3, the most appropriate kinetic
function and values of A and Ea can be determined and their validity assessed by F-test statistical
analysis. The detail of this sequential procedure was described by Hirono and Tanikawa [2011].
A two-step reaction was considered in our kinetic analysis, because dehydroxylation of illite
(from the Tokaji Mountains of northeastern Hungary) [Gualtieri and Ferrari, 2006] and illite–
muscovite (from the Rochester Shale, New York state, USA) [Hirono and Tanikawa, 2011]
showed such a reaction. The two-step reaction mechanism is expressed as
æ -E ö
da1st
(4)
= A1st exp ç a_1st ÷ f (a1st ) ,
dt
è RT ø
æ -Ea _ 2nd ö
æ -Ea _1st ö
da 2nd
(5)
= A2nd expç
÷ f (a 2nd ) - A1st expç
÷ f (a1st ),
dt
è RT ø
è RT ø
where 1st is the reacted fraction of an intermediate product at the first step (0 ≤ 1st ≤ 1) and A1st,
Ea_1st, and f(1st are the pre-exponential term, activation energy, and kinetic function,
respectively, for the first-step reaction; and 2nd is the reacted fraction of the final product after
the second step (0 ≤ 2nd ≤ 1), and A2nd, Ea_2nd, and f(2nd are the pre-exponential term,
activation energy, and kinetic function, respectively, for the second-step reaction. Using Netzsch
Thermokinetics software, including F-test statistical analysis, the most appropriate kinetic
1
function and values of A and Ea parameters can be determined. The details of this analysis have
also been documented by Hirono and Tanikawa [2011].
The best-fitted function of both first- and second-step reactions, by this analysis, was threedimensional diffusion. Values of Ea_1st and Ea_2nd determined are shown in table A1.
There are some problems that remain to be solved before we can meaningfully consider the
results of our rotary shear experiments, kinetic analyses, and the feedback processes we
considered in terms of natural faults and earthquakes.
1) The temperature reached by the highly damaged sample after 15 m of frictional slip at 1.5
MPa normal stress (Figure 1a) was higher than 300 °C, indicating that the dehydroxylation
reaction had already started. In fact, the second-step weight loss of the sample (3.72 wt%) was
0.19 wt% lower than that of the initial sample (3.91 wt%), indicating that approximately 5% of
the reaction had already occurred. Thus, there is uncertainty about the accuracy of the Ea_2nd
value determined under experimental conditions with high frictional heat such as ours.
2) The Ea_2nd value of 142.3 kJ mol-1 for our initial illite sample was lower than the 230 kJ
mol-1 value obtained for the illite of Gualtieri and Ferrari [2006] and that of 181.8 kJ mol-1 for
the illite–muscovite of Hirono and Tanikawa [2011]. These discrepancies may reflect different
polytypes and interlayer charges of the illite samples used, but this needs to be clarified.
3) The maximum frictional energy density attained in our rotary shear experiments was 17.65
MJ m-3 (for 15 m displacement at 1.5 MPa normal stress). This value corresponds to a fault slip
of only 4 cm at 1.5 km depth, assuming a slip zone that is 1 cm thick, a frictional coefficient of
0.2, hydrostatic confining stress (normal stress on the slip zone = overburden stress), and not
accounting for thermal pressurization (other parameters are shown in table A2). Thus, natural
fault slip is accompanied by much higher frictional energy density in the slip zone, probably
inducing a large decrease of Ea and causing dramatic coseismic mechanochemical effects.
However, it is difficult to demonstrate such high frictional energy density with our rotary shear
apparatus because of the likelihood of leakage of the sample and the high temperatures caused by
friction.
2
2. Dynamic Numerical Analysis of Earthquake Slip
According to heat balance, the temperature (T) at a distance x from the centre of the slip zone
is expressed in terms of the balance at time t as
¶T
¶ æ ¶T ö
(6)
( rc)e = Ah - çè le ÷ø ,
¶t
¶x
¶x
where Ah is heat production rate, (c)e is equivalent specific heat capacity,  is density, c is
specific heat, and λe is equivalent thermal conductivity. The equivalent specific heat capacity and
thermal conductivity are described respectively as
(7)
( rc)e = (1- f ) rr cr + fr f cf
and
(8)
le = (1- f ) lr + fl f ,
where  is porosity; r, cr, and r are respectively the density, specific heat capacity, and thermal
conductivity for the grain matrix (illite); and f, cf, and f are the same for pore fluid (water).
Thermal conductivity was calculated from specific heat and thermal diffusivity, , as
 = c.
(9)
Eq. 8 corresponds to an arithmetic mean that represents the upper limit of thermal conductivity
for porous media [Abdulagatova et al., 2009].
A fault model in which the slip zone is localized within a finite thickness, w, is considered
here. The strain rate is assumed to be constant across the thickness of the fault, according to Mair
and Marone [2000] and Fialko [2004]. With the assumptions that total frictional work is
converted to heat and that energy is consumed by chemical reaction, the rate of heat production
is expressed as
ù
¶é
d
v ¶E
Ah = êm ( Pc - Pf ) - Ec ú = m ( Pc - Pf ) - c (x<|w/2|)
û
¶t ë
w
w ¶t
¶E
(x|w/2|),
(10)
Ah = - c
¶t
where  is the coefficient of friction, Pc is confining pressure, Pf is pore-fluid pressure, d is slip
displacement,  is slip velocity, and Ec is the energy taken up by the endothermic reaction. Ec is
related to chemical kinetics as
(11)
Ec = arh H ,
where  is the reacted fraction, h is the density of the reactant (here hydrated mineral, illite,
=r), and H is the heat per unit mass of an endothermic reaction, which corresponds to its
enthalpy. The enthalpy is negative (H < 0) for an exothermic reaction.
The fluid diffusion equation is
¶ Pf ¶ æ k ¶ Pf ö
¶T
(12)
Ss
= ç
÷ + f (a f - a r ) + Qdeh ,
¶t ¶ x è h ¶ x ø
¶t
where Ss is the specific storage,  f is the coefficient of thermal expansion of water,  r is the
coefficient of thermal expansion of the grain matrix, Qdeh is the pore pressure generated by the
dehydroxylation reaction, k is permeability, and  is fluid viscosity. The viscosity of water is
expressed as a function of temperature as [Fontaine et al., 2001]:
h = 2.414 ´10 ´ 10
-5
247.8
T +133
.
(13)
3
Assuming that in the reaction 1 mol of hydrated mineral (illite) produces n mol of water and 1
mol of dehydrated mineral,
¶f æ n×Vf Vdeh ö
(14)
Qdeh = h ç +1÷,
¶ t è Vh
Vh
ø
where h is the volume fraction of the hydrated mineral, Vf /Vh is the molecular volume ratio of
the hydrated mineral to water, and Vdeh/Vh is the molecular volume ratio of hydrated to
dehydrated minerals [Tanikawa et al., 2009]. The chemical conversion rate is related to the
volume fraction rate as
¶a
1 ¶fh
,
(15)
=¶t
fh0 ¶ t
where /t is the same as in Eq. 1 and h0 is the initial volume fraction of the hydrated mineral.
Combining Eqs. 14 and 15, Qdeh is expressed as
¶a æ n × Vf Vdeh ö
(16)
Qdeh = -f h0
+ 1÷ .
ç¶t è Vh
Vh
ø
The molecular volume ratio, n∙Vf/Vh, in Eq. 16 was not evaluated directly because the number of
moles of water dehydrated from hydrated mineral, n, and the molecular volume of hydrated
mineral, Vh, were not certain. However, this term is expressed by the weight mass ratio in
chemical reactions as
n×Vf
1 wf rh
,
(17)
=
Vh
M f wh r f
where Mf is the molar mass of water and wf/wh is the mass ratio of hydrated mineral to water for
the dehydration reaction. The ratio wf/wh is evaluated from the average weight loss during heat
treatment of the illite sample by TG. The dehydration volume ratio, Vdeh/Vh, is evaluated from the
weight loss by thermal treatment using
Vdeh wdeh rh
,
(18)
=
Vh
wh rdeh
where wdeh/wh is the mass ratio of hydrated to dehydrated minerals and deh is the density of
dehydrated mineral. We assumed that the slip zone and its surroundings are composed of the
same illite; that is, the hydraulic, thermal, and chemical kinetic properties of the host rock
outside the slip zone were the same as those of the slip zone. In addition, the initial illite and
dehydroxylated illite samples were assigned the same physical properties (h = deh).
For the fault gouge simulation, the dependency of effective vertical pressure, Pe, on porosity
is approximated as [Wibberley and Shimamoto, 2005]
(19)
f = 0.056exp(0.00297Pe ).
For illite-rich clay, the dependency of effective vertical pressure on the void ratio, evoid, is
approximated as [Djeran-Maigre et al., 1998]
(20)
evoid = 0.66 - 0.32logPe.
The void ratio can be converted to porosity using
1
.
(21)
f=
evoid +1
The specific storage, Ss, in Eq. 12 is calculated from the porosity and fluid compressibility as
[Tanikawa et al., 2008]
4
1 df
(22)
+ bf ,
1- f dPe
where  is the compressibility of water. Hydraulic diffusivity, Dh, was calculated as
k
.
(23)
Dh =
hSs
For reference, the change of Dh with Pe from 2 to 100 MPa at temperatures of 300 and 1300 K is
shown in figure A3.
By using the finite difference method (time increment of 0.001 s and grid size of 0.5 mm),
these equations (Eqs. 4–6, 11–23) were simultaneously solved with assumptions of 100 wt%
illite composition and hydrostatic confining stress (normal stress acting on the slip zone =
overburden stress), and the changes with time of pore-fluid pressure, shear stress, temperature,
and reacted fraction at position x were obtained. Values of the parameters required for this
simulation are summarized in table A2. Since Pf reaches Pc, we assumed that temperature
remains constant and energy taken up by the dehydroxylation reaction is ignored.
The resultant profiles of Pf, shear stress, 2nd, T, and hydraulic diffusivity with slip
displacement are discussed in the main text (Figure 3). Additionally, we note here the following.
1) Although two damaged cases of 1.5 and 6.0 km depths were applied using only Ea = 97.0
mol-1, the frictional energy densities at such depths can immediately reach 17.65 MJ m-3, thus
allowing Ea = 97.0 kJ mol-1 just after the start of slip.
2) Hydraulic diffusivity for the fault gouge of Wibberley and Shimamoto [2005] is lower than
that of the illite-rich clay of Djeran-Maigre et al. [1998] for Pe < ~20 MPa (figure A3), which
strongly affects the intense pressurization of the interstitial fluids and H2O released from the illite
sample in the fault gouge of Wibberley and Shimamoto [2005] (Figure 3a).
2) Changes of pore-fluid pressure in the fault gouge of Wibberley and Shimamoto [2005]
showed a two-step increase (Figure 3a), the first corresponding to pressurization of interstitial
fluid and the second to release of H2O from the illite sample (Figure 3a, e).
4) After Pf reached Pc, the reacted fractions increased with time because of the assumed
constant temperature (Figure 3e, g). However, in natural faults, the heat remaining after slip
could lengthen the time of reaction [e.g., Hirono et al., 2008; Hamada et al., 2009].
5) The temperature at which dry illite melts (Figure 3g, h) was assumed to be approximately
1150 °C [Balkyavichus et al., 2003], but wet conditions in a natural setting may lower that
temperature. Thus, melt lubrication may take effect dominantly under the conditions of higher
shear stress at greater depths.
6) Although only 1-cm-thick slip zone was applied for our simulation, not only permeability
and mechanochemical effect but also the thickness strongly affects numerical results [e.g.,
Hirono and Tanikawa, 2011].
7) The pressure dependency of kinetic parameters (Ea) was not considered in our simulation.
In addition, faster heating during earthquake slip may require different kinetic functions and
parameters than those for heating at 5–20 °C min–1. Future work should attempt to determine
these parameters under high-pressure and at fast rates of heating to provide a more realistic
evaluation of fault behavior.
Ss =
5
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